Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of t...

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of t...

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of t...

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of t...

I consider an initial value problem for the offset continuation (OC) equation introduced in Part One of this paper (SEP–84). The solutions of this problem create integral-type OC operators in the time-space domain. Moving to the frequency-wavenumber and log-stretch domain, I compare the obtained operators with the well-known Fourier DMO operators. This comparison links the theory of DMO with th...

We consider an obstacle-type problem ∆u = f(x)χΩ in D u = |∇u| = 0 on D \ Ω, where D is a given open set in Rn and Ω is an unknown open subset of D. The problem originates in potential theory, in connection with harmonic continuation of potentials. The qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign. Using geom...

In this article we study the existence, continuation and bifurcation from infinity of nonconstant solutions for a nonlinear Neumann problem. We apply the LeraySchauder degree and the degree for SO(2)-equivariant gradient operators defined by the second author in [21].

Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. Continuation MPC, suggested by T. Ohtsuka in 2004, uses Krylov-Newton iterations. Continuation MPC is suitable for nonlinear problems and has been recently adopted for minimum time problems. We extend the continuation MPC ...

Recently, a couple of approaches have been developed that combine multiobjective optimization with direct discretization methods to approximate trajectories of optimal control problems, resulting in restricted optimization problems of high dimension. The solution set of a multiobjective optimization problem is called the Pareto set which consists of optimal compromise solutions. A common way to...